Tapered thermal actuator

ABSTRACT

An apparatus for a thermal actuator for a micromechanical device, especially a liquid drop emitter such as an ink jet printhead, is disclosed. The disclosed thermal actuator comprises a base element and a cantilevered element including a thermo-mechanical bending portion extending from the base element and a free end portion residing in a first position. The thermo-mechanical bending portion has a base end width, w b , adjacent the base element and a free end width, w f , adjacent the free end portion wherein the base end width is substantially greater than the free end width. The thermal actuator further comprises apparatus adapted to apply a heat pulse directly to the thermo-mechanical bending portion causing the deflection of the free end portion of the cantilevered element to a second position. The width of the thermo-mechanical bending portion may reduce substantially quadratically or in an inverse power fashion as a function of the distance away from the base element or in at least one step reduction. The apparatus adapted to apply a heat pulse may comprise a thin film resistor. Alternatively, the thermo-mechanical bending portion may comprise a layer of electrically resistive material having a heater resistor formed therein to which is applied an electrical pulse to cause rapid deflection of the free end portion and ejection of a liquid drop.

CROSS REFERENCE TO RELATED APPLICATION

[0001] This is a continuation-in-part of commonly assigned U.S.application Ser. No. 10/227,079, entitled “Tapered Thermal Actuator,”filed Aug. 23, 2002.

FIELD OF THE INVENTION

[0002] The present invention relates generally tomicro-electromechanical devices and, more particularly, tomicro-electromechanical thermal actuators such as the type used in inkjet devices and other liquid drop emitters.

BACKGROUND OF THE INVENTION

[0003] Micro-electro mechanical systems (MEMS) are a relatively recentdevelopment. Such MEMS are being used as alternatives to conventionalelectro-mechanical devices as actuators, valves, and positioners.Micro-electromechanical devices are potentially low cost, due to use ofmicroelectronic fabrication techniques. Novel applications are alsobeing discovered due to the small size scale of MEMS devices.

[0004] Many potential applications of MEMS technology utilize thermalactuation to provide the motion needed in such devices. For example,many actuators, valves and positioners use thermal actuators formovement. In some applications the movement required is pulsed. Forexample, rapid displacement from a first position to a second, followedby restoration of the actuator to the first position, might be used togenerate pressure pulses in a fluid or to advance a mechanism one unitof distance or rotation per actuation pulse. Drop-on-demand liquid dropemitters use discrete pressure pulses to eject discrete amounts ofliquid from a nozzle.

[0005] Drop-on-demand (DOD) liquid emission devices have been known asink printing devices in ink jet printing systems for many years. Earlydevices were based on piezoelectric actuators such as are disclosed byKyser et al., in U.S. Pat. No. 3,946,398 and Stemme in U.S. Pat. No.3,747,120. A currently popular form of ink jet printing, thermal ink jet(or “bubble jet”), uses electroresistive heaters to generate vaporbubbles which cause drop emission, as is discussed by Hara et al., inU.S. Pat. No. 4,296,421.

[0006] Electroresistive heater actuators have manufacturing costadvantages over piezoelectric actuators because they can be fabricatedusing well developed microelectronic processes. On the other hand, thethermal ink jet drop ejection mechanism requires the ink to have avaporizable component, and locally raises ink temperatures well abovethe boiling point of this component. This temperature exposure placessevere limits on the formulation of inks and other liquids that may bereliably emitted by thermal ink jet devices. Piezoelectrically actuateddevices do not impose such severe limitations on the liquids that can bejetted because the liquid is mechanically pressurized.

[0007] The availability, cost, and technical performance improvementsthat have been realized by ink jet device suppliers have also engenderedinterest in the devices for other applications requiring micro-meteringof liquids. These new applications include dispensing specializedchemicals for micro-analytic chemistry as disclosed by Pease et al., inU.S. Pat. No. 5,599,695; dispensing coating materials for electronicdevice manufacturing as disclosed by Naka et al., in U.S. Pat. No.5,902,648; and for dispensing microdrops for medical inhalation therapyas disclosed by Psaros et al., in U.S. Pat. No. 5,771,882. Devices andmethods capable of emitting, on demand, micron-sized drops of a broadrange of liquids are needed for highest quality image printing, but alsofor emerging applications where liquid dispensing requiresmono-dispersion of ultra small drops, accurate placement and timing, andminute increments.

[0008] A low cost approach to micro drop emission is needed which can beused with a broad range of liquid formulations. Apparatus and methodsare needed which combine the advantages of microelectronic fabricationused for thermal ink jet with the liquid composition latitude availableto piezo-electromechanical devices.

[0009] A DOD ink jet device which uses a thermo-mechanical actuator wasdisclosed by T. Kitahara in JP 2,030,543, filed Jul. 21, 1988. Theactuator is configured as a bi-layer cantilever moveable within an inkjet chamber. The beam is heated by a resistor causing it to bend due toa mismatch in thermal expansion of the layers. The free end of the beammoves to pressurize the ink at the nozzle causing drop emission.Recently, disclosures of a similar thermo-mechanical DOD ink jetconfiguration have been made by K. Silverbrook in U.S. Pat. Nos.6,067,797; 6,087,638; 6,239,821 and 6,243,113. Methods of manufacturingthermo-mechanical ink jet devices using microelectronic processes havebeen disclosed by K. Silverbrook in U.S. Pat. Nos. 6,180,427; 6,254,793and 6,274,056.

[0010] Thermo-mechanically actuated drop emitters employing acantilevered element are promising as low cost devices which can be massproduced using microelectronic materials and equipment and which allowoperation with liquids that would be unreliable in a thermal ink jetdevice. However, the design and operation of cantilever style thermalactuators and drop emitters requires careful attention to energyefficiency so as to manage peak temperature excursions and maximizeactuation repetition frequencies. Designs which produce a comparableamount of deflection and a deflection force while requiring less inputenergy than previous configurations are needed to enhance the commercialpotential of various thermally actuated devices, especially ink jetprintheads.

[0011] Configurations for cantilevered element thermal actuators,optimized for input energy efficiency, are needed which can be operatedat high repetition frequencies and with maximum force of actuation.

SUMMARY OF THE INVENTION

[0012] It is therefore an object of the present invention to provide athermo-mechanical actuator which operates with improved energyefficiency.

[0013] It is also an object of the present invention to provide a liquiddrop emitter which operates with improved energy efficiency.

[0014] The foregoing and numerous other features, objects and advantagesof the present invention will become readily apparent upon a review ofthe detailed description, claims and drawings set forth herein. Thesefeatures, objects and advantages are accomplished by constructing athermal actuator for a micro-electromechanical device comprising a baseelement and a cantilevered element which includes a thermo-mechanicalbending portion extending from the base element and a free end portionresiding in a first position. The thermo-mechanical bending portion hasa base end width, w_(b), adjacent the base element and a free end width,w_(f), adjacent the free end portion wherein the base end width issubstantially greater than the free end width. The thermal actuatorfurther comprises apparatus adapted to apply a heat pulse directly tothe thermo-mechanical bending portion causing the deflection of the freeend portion of the cantilevered element to a second position. The widthof the thermo-mechanical bending portion may reduce as a function of thedistance away from the base element in a functional form that results ina normalized deflection of the free end {overscore (y)}(1)<1.0. Theapparatus adapted to apply a heat pulse may comprise a thin filmresistor. Alternatively, the thermo-mechanical bending portion maycomprise a first layer of an electrically resistive material having aheater resistor formed therein to which is applied an electrical pulsethereby causing rapid deflection of the free end portion.

[0015] The present invention is particularly useful as a thermalactuator for liquid drop emitters used as printheads for DOD ink jetprinting. In this preferred embodiment the thermal actuator resides in aliquid-filled chamber that includes a nozzle for ejecting liquid. Thethermal actuator includes a cantilevered element extending from a wallof the chamber and a free end residing in a first position proximate tothe nozzle. Application of a heat pulse to the cantilevered elementcauses deflection of the free end forcing liquid from the nozzle.

BRIEF DESCRIPTION OF THE DRAWINGS

[0016]FIG. 1 is a schematic illustration of an ink jet system accordingto the present invention;

[0017]FIG. 2 is a plan view of an array of ink jet units or liquid dropemitter units according to the present invention;

[0018] FIGS. 3(a) and 3(b) are enlarged plan views of an individual inkjet unit shown in FIG. 2;

[0019] FIGS. 4(a) and 4(b) are side views illustrating the movement of athermal actuator according to the present invention;

[0020]FIG. 5 is a perspective view of the early stages of a processsuitable for constructing a thermal actuator according to the presentinvention wherein a first layer of electrically resistive material ofthe cantilevered element is formed;

[0021]FIG. 6 is a perspective view of a next stage of the processillustrated in FIG. 5 wherein a current coupling device is added;

[0022]FIG. 7 is a perspective view of the next stages of the processillustrated in FIGS. 5 or 6 wherein a second layer of a dielectricmaterial of the cantilevered element is formed;

[0023]FIG. 8 is a perspective view of the next stages of the processillustrated in FIGS. 5-7 wherein a sacrificial layer in the shape of theliquid filling a chamber of a drop emitter according to the presentinvention is formed;

[0024]FIG. 9 is a perspective view of the next stages of the processillustrated in FIGS. 5-8 wherein a liquid chamber and nozzle of a dropemitter according to the present invention is formed;

[0025] FIGS. 10(a)-10(c) are side views of the final stages of theprocess illustrated in FIGS. 5-9 wherein a liquid supply pathway isformed and the sacrificial layer is removed to complete a liquid dropemitter according to the present invention;

[0026] FIGS. 11(a) and 11(b) are side views illustrating the operationof a drop emitter according the present invention;

[0027] FIGS. 12(a) and (b) are plan views of alternative designs for athermo-mechanical bending portion according to the present inventions;

[0028] FIGS. 13(a) and 13(b) are perspective and plan views of a designfor a thermo-mechanical bending portion according to the presentinventions;

[0029]FIG. 14 is a plot of thermo-mechanical bending portion free enddeflection under an imposed load for tapered thermo-mechanical actuatorsas a function of taper angle;

[0030] FIGS. 15(a)-15(c) are plan views of alternative designs for athermo-mechanical bending portion according to the present inventions;

[0031]FIG. 16 is a plot of thermo-mechanical bending portion free enddeflection under an imposed load for stepped reduction thermo-mechanicalactuators as a function of width reduction;

[0032]FIG. 17 is a plot of the parameters of a single step reductionshaped thermo-mechanical bender portion that yield the minimumnormalized deflection of the free end;

[0033]FIG. 18 is a plot of the minimum normalized deflection of the freeend of a single step reduction thermo-mechanical bender portionresulting from the optimum parameters plotted in FIG. 17, as a functionof the step position;

[0034]FIG. 19 shows contour plots of the thermo-mechanical bendingportion free end deflection under an imposed load for single stepreduction thermo-mechanical actuators as a function of step position andfree end width reduction;

[0035] FIGS. 20(a) and 20(b) are plan views of alternative designs for athermo-mechanical bending portion according to the present inventions;

[0036]FIG. 21 shows contour plots of the thermo-mechanical bendingportion free end deflection under an imposed load for width reductionshapes of the form illustrated in FIG. 20;

[0037] FIGS. 22(a)-22(c) are plan views of alternative designs for athermo-mechanical bending portion;

[0038]FIG. 23 shows contour plots of the thermo-mechanical bendingportion free end deflection under an imposed load for width reductionshapes of the form illustrated in FIG. 22;

[0039]FIG. 24 plots a numerical simulation of the peak deflection of atapered thermo-mechanical actuator, when actuated, as a function oftaper angle.

DETAILED DESCRIPTION OF THE INVENTION

[0040] The invention has been described in detail with particularreference to certain preferred embodiments thereof, but it will beunderstood that variations and modifications can be effected within thespirit and scope of the invention.

[0041] As described in detail herein below, the present inventionprovides apparatus for a thermal actuator and a drop-on-demand liquidemission device. The most familiar of such devices are used asprintheads in ink jet printing systems. Many other applications areemerging which make use of devices similar to ink jet printheads,however which emit liquids other than inks that need to be finelymetered and deposited with high spatial precision. The terms ink jet andliquid drop emitter will be used herein interchangeably. The inventionsdescribed below provide drop emitters based on thermo-mechanicalactuators which are configured and operated so as to avoid locations ofexcessive temperature, hot spots, which might otherwise cause erraticperformance and early device failure.

[0042] Turning first to FIG. 1, there is shown a schematicrepresentation of an ink jet printing system which may use an apparatusand be operated according to the present invention. The system includesan image data source 400 which provides signals that are received bycontroller 300 as commands to print drops. Controller 300 outputssignals to a source of electrical pulses 200. Pulse source 200, in turn,generates an electrical voltage signal composed of electrical energypulses which are applied to electrically resistive means associated witheach thermo-mechanical actuator 15 within ink jet printhead 100. Theelectrical energy pulses cause a thermo-mechanical actuator 15 (hereinafter “thermal actuator”) to rapidly bend, pressurizing ink 60 locatedat nozzle 30, and emitting an ink drop 50 which lands on receiver 500.

[0043]FIG. 2 shows a plan view of a portion of ink jet printhead 100. Anarray of thermally actuated ink jet units 110 is shown having nozzles 30centrally aligned, and ink chambers 12, interdigitated in two rows. Theink jet units 110 are formed on and in a substrate 10 usingmicroelectronic fabrication methods. An example fabrication sequencewhich may be used to form drop emitters 110 is described in co-pendingapplication Ser. No. 09/726,945 filed Nov. 30, 2000, for “ThermalActuator”, assigned to the assignee of the present invention.

[0044] Each drop emitter unit 110 has associated electrical leadcontacts 42, 44 which are formed with, or are electrically connected to,a heater resistor portion 25, shown in phantom view in FIG. 2. In theillustrated embodiment, the heater resistor portion 25 is formed in afirst layer of a cantilevered element 20 of a thermal actuator andparticipates in the thermo-mechanical effects as will be described.Element 80 of the printhead 100 is a mounting structure which provides amounting surface for microelectronic substrate 10 and other means forinterconnecting the liquid supply, electrical signals, and mechanicalinterface features.

[0045]FIG. 3a illustrates a plan view of a single drop emitter unit 110and a second plan view FIG. 3b with the liquid chamber cover 28,including nozzle 30, removed.

[0046] The thermal actuator 15, shown in phantom in FIG. 3a can be seenwith solid lines in FIG. 3b. The cantilevered element 20 of thermalactuator 15 extends from base element edge 14 of liquid chamber 12 whichis formed in substrate base element 10. Cantilevered element anchorportion 26 is bonded to base element substrate 10 and anchors thecantilever.

[0047] The cantilevered element 20 of the actuator has the shape of apaddle, an extended, tapered flat shaft ending with a disc of largerdiameter than the final shaft width. This shape is merely illustrativeof cantilever actuators which can be used, many other shapes areapplicable as will be described hereinbelow. The disc-shape aligns thenozzle 30 with the center of the cantilevered element free end portion27. The fluid chamber 12 has a curved wall portion at 16 which conformsto the curvature of the free end portion 27, spaced away to provideclearance for the actuator movement.

[0048]FIG. 3b illustrates schematically the attachment of electricalpulse source 200 to the resistive heater 25 at interconnect terminals 42and 44. Voltage differences are applied to voltage terminals 42 and 44to cause resistance heating via heater resistor 25. This is generallyindicated by an arrow showing a current I. In the plan views of FIG. 3,the actuator free end portion 27 moves toward the viewer when pulsed anddrops are emitted toward the viewer from the nozzle 30 in cover 28. Thisgeometry of actuation and drop emission is called a “roof shooter” inmany ink jet disclosures.

[0049]FIG. 4 illustrates in side view a cantilevered thermal actuator 15according to a preferred embodiment of the present invention. In FIG. 4athe actuator is in a first position and in FIG. 4b it is shown deflectedupward to a second position. Cantilevered element 20 extends from ananchor location 14 of base element 10. The cantilevered element 20 isconstructed of several layers. First layer 22 causes the upwarddeflection when it is thermally elongated with respect to other layersin the cantilevered element 20. It is constructed of an electricallyresistive material, preferably intermetallic titanium aluminide, thathas a large coefficient of thermal expansion.

[0050] A current coupling device 68 is illustrated in side view in FIG.4. The current coupling device conducts current serially between twoelongated resistor segments of heater resistor 25 and may be formed bydepositing and patterning a metallic layer such as aluminum or by usingthe electrically resistive material.

[0051] The cantilevered element 20 also includes a second layer 23,attached to the first layer 22. The second layer 23 is constructed of asecond material having a low coefficient of thermal expansion, withrespect to the material used to construct the first layer 22. Thethickness of second layer 23 is chosen to provide the desired mechanicalstiffness and to maximize the deflection of the cantilevered element fora given input of heat energy. Second layer 23 may also be a dielectricinsulator to provide electrical insulation for resistive heater segmentsand current coupling devices and segments formed into the first layer orin a third material used in some preferred embodiments of the presentinventions. The second layer may be used to partially defineelectroresistor and current coupler devices formed as portions of firstlayer 22 or in an added conductive layer.

[0052] Second layer 23 may be composed of sub-layers, laminations ofmore than one material, so as to allow optimization of functions of heatflow management, electrical isolation, and strong bonding of the layersof the cantilevered element 20.

[0053] Passivation layer 21 shown in FIG. 4 is provided to protect thefirst layer 22 chemically and electrically. Such protection may not beneeded for some applications of thermal actuators according to thepresent invention, in which case it may be deleted. Liquid drop emittersutilizing thermal actuators which are touched on one or more surfaces bythe working liquid may require passivation layer 21 which is chemicallyand electrically inert to the working liquid.

[0054] The overall thickness, h, of cantilevered element 20 is indicatedin FIG. 4. In the immediate area of current coupling device 68 it may besomewhat thicker if an added material is used to form the currentcoupler.

[0055] A heat pulse is applied to first layer 22, causing it to rise intemperature and elongate. Second layer 23 does not elongate nearly asmuch because of its smaller coefficient of thermal expansion and thetime required for heat to diffuse from first layer 22 into second layer23. The difference in length between first layer 22 and the second layer23 causes the cantilevered element 20 to bend upward an amount D, asillustrated in FIG. 4b. When used as an actuator in a drop emitter, thebending response of the cantilevered element 20 must be rapid enough tosufficiently pressurize the liquid at the nozzle. Typically,electroresistive heating apparatus is adapted to apply heat pulses, andan electrical pulse duration of less than 4 μsecs. is used and,preferably, a duration less than 2 μsecs.

[0056]FIGS. 5 through 10 illustrate fabrication processing steps forconstructing a single liquid drop emitter according to some of thepreferred embodiments of the present invention. For these embodimentsthe first layer 22 is constructed using an electrically resistivematerial, such as titanium aluminide, and a portion is patterned into aresistor for carrying electrical current, I.

[0057]FIG. 5 illustrates a first layer 22 of a cantilever in a firststage of fabrication. The illustrated structure is formed on a substrate10, for example, single crystal silicon, by standard microelectronicdeposition and patterning methods. A portion of substrate 10 will alsoserve as a base element from which cantilevered element 20 extends.Deposition of intermetallic titanium aluminide may be carried out, forexample, by RF or pulsed DC magnetron sputtering. An example depositionprocess that may be used for titanium aluminide is described inco-pending application Ser. No. 09/726,945 filed Nov. 30, 2000, for“Thermal Actuator”, assigned to the assignee of the present invention.

[0058] After first layer 22 is deposited it is patterned by removingmaterial to create desired shapes for thermo-mechanical performance aswell as an appropriate electrical current path for purposes of applyinga heat pulse. A cantilever free end portion 27 is illustrated.Addressing electrical leads 42 and 44 are illustrated as being formed inthe first layer 22 material as well. Leads 42, 44 may make contact withcircuitry previously formed in base element substrate 10 or may becontacted externally by other standard electrical interconnectionmethods, such as tape automated bonding (TAB) or wire bonding. Apassivation layer 21 is formed on substrate 10 before the deposition andpatterning of the first layer 22 material. This passivation layer may beleft under first layer 22 and other subsequent structures or removed ina subsequent patterning process.

[0059]FIG. 6 illustrates a next step in the fabrication processfollowing the step illustrated previously. In this step a currentcoupling device 68 is formed at the location where the free end portion27 joins the shaft of the cantilevered element. In the illustratedembodiment, the current coupling device 68 is formed by depositing andpatterning a conductive material which serially conducts current betweenelongated heater resistor segments 66. The heat pulse activation currentpath is indicated by an arrow and letter “I”. The coupler segment 68reverses the direction of current and serves to define the outer end ofthe directly heated portion of the cantilevered element.

[0060]FIG. 7 illustrates a second layer 23 having been deposited andpatterned over the previously formed first layer 22 portion of thethermal actuator. Second layer 23 also covers the current couplingdevice 68. Second layer 23 is formed over the first layer 22 coveringthe remaining resistor pattern including resistor segments 66. Thesecond layer 23 material has low coefficient of thermal expansioncompared to the material of first layer 22. For example, second layer 23may be silicon dioxide, silicon nitride, aluminum oxide or somemulti-layered lamination of these materials or the like.

[0061] In FIG. 7, a trapezoidal-shaped portion of the cantileveredelement is illustrated extending between dotted lines. The indicatedportion is a thermo-mechanical bending device comprised of high thermalexpansion layer 22 and low thermal expansion layer 23. Later, whenreleased from substrate 10, thermo-mechanical bending portion 68 willbend upward when an electrical pulse is applied to the heater resistor25 formed in first layer 22.

[0062] Additional passivation materials may be applied at this stageover the second layer 23 for chemical and electrical protection. Also,the initial passivation layer 21 is patterned away from areas throughwhich fluid will pass from openings to be etched in substrate 10.

[0063]FIG. 8 shows the addition of a sacrificial layer 29 which isformed into the shape of the interior of a chamber of a liquid dropemitter. A suitable material for this purpose is polyimide. Polyimide isapplied to the device substrate in sufficient depth to also planarizethe surface which has the topography of the first 22 and second 23layers as illustrated in FIGS. 5-7. Any material which can beselectively removed with respect to the adjacent materials may be usedto construct sacrificial structure 29.

[0064]FIG. 9 illustrates drop emitter liquid chamber walls and coverformed by depositing a conformal material, such as plasma depositedsilicon oxide, nitride, or the like, over the sacrificial layerstructure 29. This layer is patterned to form drop emitter chamber 28.Nozzle 30 is formed in the drop emitter chamber, communicating to thesacrificial material layer 29, which remains within the drop emitterchamber 28 at this stage of the fabrication sequence.

[0065]FIG. 10 shows a side view of the device through a sectionindicated as A-A in FIG. 9. In FIG. 10a the sacrificial layer 29 isenclosed within the drop emitter chamber walls 28 except for nozzleopening 30. Also illustrated in FIG. 10a, the substrate 10 is intact.Passivation layer 21 has been removed from the surface of substrate 10in gap area 13 and around the periphery of the cantilevered element 20.The removal of layer 21 in these locations was done at a fabricationstage before the forming of sacrificial structure 29.

[0066] In FIG. 10b, substrate 10 is removed beneath the cantileverelement 20 and the liquid chamber areas around and beside the cantileverelement 20. The removal may be done by an anisotropic etching processsuch as reactive ion etching, or such as orientation dependent etchingfor the case where the substrate used is single crystal silicon. Forconstructing a thermal actuator alone, the sacrificial structure andliquid chamber steps are not needed and this step of etching awaysubstrate 10 may be used to release the cantilevered element 20.

[0067] In FIG. 10c the sacrificial material layer 29 has been removed bydry etching using oxygen and fluorine sources. The etchant gasses entervia the nozzle 30 and from the newly opened fluid supply chamber area12, etched previously from the backside of substrate 10. This stepreleases the cantilevered element 20 and completes the fabrication of aliquid drop emitter structure.

[0068]FIG. 11 illustrates a side view of a liquid drop emitter structureaccording to some preferred embodiments of the present invention. FIG.11a shows the cantilevered element 20 in a first position proximate tonozzle 30. FIG. 11b illustrates the deflection of the free end 27 of thecantilevered element 20 towards nozzle 30. Rapid deflection of thecantilevered element to this second position pressurizes liquid 60causing a drop 50 to be emitted.

[0069] In an operating emitter of the cantilevered element typeillustrated, the quiescent first position may be a partially bentcondition of the cantilevered element 20 rather than the horizontalcondition illustrated FIG. 11a. The actuator may be bent upward ordownward at room temperature because of internal stresses that remainafter one or more microelectronic deposition or curing processes. Thedevice may be operated at an elevated temperature for various purposes,including thermal management design and ink property control. If so, thefirst position may be as substantially bent as is illustrated in FIG.11b.

[0070] For the purposes of the description of the present inventionherein, the cantilevered element will be said to be quiescent or in itsfirst position when the free end is not significantly changing indeflected position. For ease of understanding, the first position isdepicted as horizontal in FIG. 4a and FIG. 11a. However, operation ofthermal actuators about a bent first position are known and anticipatedby the inventors of the present invention and are fully within the scopeof the present inventions.

[0071]FIGS. 5 through 10 illustrate a preferred fabrication sequence.However, many other construction approaches may be followed using wellknown microelectronic fabrication processes and materials. For thepurposes of the present invention, any fabrication approach whichresults in a cantilevered element including a thermo-mechanical bendingportion may be followed. In addition, the thermo-mechanical bendingportion may be heated by other apparatus adapted to apply a heat pulse.For example, a thin film resistor may be formed beneath or above thethermo-mechanical bending portion and electrically pulsed to apply heat.Alternatively, heating pulses may be applied to the thermo-mechanicalbending portion by light energy or electromagnetic coupling.

[0072] In the illustrated sequence of FIGS. 5 through 10, the liquidchamber 28 and nozzle 30 of a liquid drop emitter were formed in situ onsubstrate 10. Alternatively a thermal actuator could be constructedseparately and bonded to a liquid chamber component to form a liquiddrop emitter.

[0073] The inventors of the present inventions have discovered that theefficiency of a cantilevered element thermal actuator is importantlyinfluenced by the shape of the thermal bending portion. The cantileveredelement is designed to have a length sufficient to result in an amountof deflection sufficient to meet the requirements of the microelectronicdevice application, be it a drop emitter, a switch, a valve, lightdeflector, or the like. The details of thermal expansion differences,stiffness, thickness and other factors associated with the layers of thethermo-mechanical bending portion are considered in determining anappropriate length for the cantilevered element.

[0074] The width of the cantilevered element is important in determiningthe force which is achievable during actuation. For most applications ofthermal actuators, the actuation must move a mass and overcome counterforces. For example, when used in a liquid drop emitter, the thermalactuator must accelerate a mass of liquid and overcome backpressureforces in order to generate a pressure pulse sufficient to emit a drop.When used in switches and valves the actuator must compress materials toachieve good contact or sealing.

[0075] In general, for a given length and material layer construction,the force that may be generated is proportional to the width of thethermo-mechanical bending portion of the cantilevered element. Astraightforward design for a thermo-mechanical bender is therefore arectangular beam of width w₀ and length L, wherein L is selected toproduce adequate actuator deflection and w₀ is selected to produceadequate force of actuation, for a given set of thermo-mechanicalmaterials and layer constructions.

[0076] It has been found by the inventors of the present inventions thatthe straightforward rectangular shape mentioned above is not the mostenergy efficient shape for the thermo-mechanical bender. Rather, it hasbeen discovered that a thermo-mechanical bending portion that reduces inwidth from the anchored end of the cantilevered element to a narrowerwidth at the free end, produces more force for a given area of thebender.

[0077]FIG. 12a illustrates a cantilevered element 27 andthermo-mechanical bending portion 63 according to the present invention.Thermo-mechanical bending portion 63 extends from the base elementanchor location 14 to a location of connection 18 to free end portion27. The width of the thermo-mechanical bending portion is substantiallygreater at the base end, w_(b), than at the free end, w_(f). In FIG.12a, the width of the thermo-mechanical bender decreases linearly fromw_(b) to w_(f) producing a trapezoidal shaped thermo-mechanical bendingportion. Also illustrated in FIG. 12a, w_(b) and w_(f) are chosen sothat the area of the trapezoidal thermo-mechanical bending portion 63,is equal to the area of a rectangular thermo-mechanical bending portion,shown in phantom in FIG. 12a, having the same length L and a widthw₀=½(w_(b)+w_(f)).

[0078] The linear tapering shape illustrated in FIG. 12a is a specialcase of a generally tapering shape according to the present inventionsand illustrated in FIG. 12b. Generally tapering thermo-mechanicalbending portion 62, illustrated in FIG. 12b, has a width, w(x), whichdecreases monotonically as a function of the distance, x, from w_(b) atanchor location 14 at base element 10, to w_(f) at the location ofconnection 18 to free end portion 27 at distance L. In FIG. 12b, thedistance variable x, over which the thermo-mechanical bending portion 62monotonically reduces in width, is expressed as covering a range x=0→1,i.e. in units normalized by length L.

[0079] The beneficial effect of a taper-shaped thermo-mechanical bendingportion 62 or 63 may be understood by analyzing the resistance tobending of a beam having such a shape. FIG. 13 illustrates a first shapethat can be explored analytically in closed form. FIG. 13 a shows inperspective view a cantilevered element 20 comprised of first and secondlayers 22 and 23. A linearly-tapered (trapezoidal) thermo-mechanicalbending portion 63 extends from anchor location 14 of base element 10 toa free end portion 27. A force, P, representing a load or backpressure,is applied perpendicularly, in the negative y-direction in FIG. 13, tothe free end 18 of thermo-mechanical bending portion 63 where it joinsto free end portion 27 of the cantilevered element.

[0080]FIG. 13b illustrates in plan view the geometrical features of atrapezoidal thermo-mechanical bending portion 63 that are used in theanalysis hereinbelow. Note that the amount of linear taper is expressedas an angle Θ in FIG. 13b and as a difference width, δw₀/2, in FIG. 12b.These two descriptions of the taper are related as follows: tan Θ=δw₀/L.

[0081] Thermo-mechanical bending portion 63, fixed at anchor location 14(x=0) and impinged by force P at free end 18 (x=L) assumes anequilibrium shape based on geometrical parameters, including the overallthickness h, and the effective Young's modulus, E, of the multi-layerstructure. The anchor connection exerts a force, oppositely directed tothe force P, on the cantilevered element that prevents it fromtranslating. Therefore the net moment, M(x), acting on thethermo-mechanical bending portion at a distance, x from the fixed baseend is:

M(x)=Px−PL.  (1)

[0082] The thermo-mechanical bending portion 63 resists bending inresponse to the applied moment, M(x), according to geometrical shapefactors expressed as the beam stiffness I(x) and Young's modulus, E.Therefore: $\begin{matrix}{{{{{EI}(x)}\frac{^{2}y}{x^{2}}} = {M(x)}},{where}} & (2) \\{{I(x)} = {{\frac{1}{12}{w(x)}{h^{3}.\quad {Combining}}\quad {with}\quad {{Eq}.\quad 1}}:}} & (3) \\{\frac{^{2}y}{x^{2}} = {\frac{12{PL}^{3}}{{Eh}^{3}}{\frac{\left( {x - 1} \right)}{w(x)}.}}} & (4)\end{matrix}$

[0083] Equation 4 above is a differential equation in y(x), thedeflection of the thermo-mechanical bending member as a function of thegeometrical parameters, materials parameters and distance out from thefixed anchor location, x, expressed in units of L. Equation 4 may besolved for y(x) using the boundary conditions y(0)=dy(0)/dx=0.

[0084] It is useful to solve Equation 4 initially for a rectangularthermo-mechanical bending portion to establish a base or nominal casefor comparison to the reducing width shapes of the present inventions.Thus, for the rectangular shape illustrated in phantom lines in FIG.12a, $\begin{matrix}{{{w(x)} = w_{0}},{0 \leq x \leq 1.0},} & (5) \\{{\frac{^{2}y}{x^{2}} = {\frac{12{PL}^{3}}{{Eh}^{3}}\frac{\left( {x - 1} \right)}{w_{0}}}},} & (6) \\{{{y(x)} = {C_{0}\left( {\frac{x^{3}}{6} - \frac{x^{2}}{2}} \right)}},{where},} & (7) \\{C_{0} = {\frac{12{PL}^{3}}{{Eh}^{3}w_{0}}.}} & (8)\end{matrix}$

[0085] At the free end of the rectangular thermo-mechanical bendingportion 63, x=1.0, the deflection of the beam, y(1), in response to aload P is therefore: $\begin{matrix}{{y(1)} = {{- \frac{1}{3}}{C_{0}.}}} & (9)\end{matrix}$

[0086] The deflection of the free end location 18 of a rectangularthermo-mechanical bending portion, y(1), expressed in above Equation 9,will be used in the analysis hereinbelow as a normalization factor. Thatis, the amount of deflection under a load P of thermo-mechanical bendingportions having reducing widths according to the present inventions,will be analyzed and compared to the rectangular case. It will be shownthat the thermo-mechanical bending portions of the present inventionsare deflected less by an equal load or backpressure than a rectangularthermo-mechanical bending portion having the same length, L, and averagewidth, w₀. Because the shapes of the thermo-mechanical bending portionsaccording to the present inventions are more resistant to load forcesand backpressure forces, more deflection and more forceful deflectioncan be achieved by the input of the same heat energy as compared to arectangular thermo-mechanical bender.

[0087] Trapezoidal-shaped thermo-mechanical bending portions, asillustrated in FIGS. 2, 3, 12, and 13 are preferred embodiments of thepresent inventions. The thermo-mechanical bending portion 63 is designedto narrow from a base end width, w_(b), to a free end width, w_(f), in alinear function of x, the distance out from the anchor location 14 ofbase element 10. Further, to clarify the improved efficiencies that areobtained, the trapezoidal-shaped thermo-mechanical bending portion isdesigned to have the same length, L, and area, w₀L, as therectangular-shaped thermo-mechanical bending portion described by aboveEquation 5. The trapezoidal-shape width function, w(x), may be expressedas:

w(x)=w ₀(ax+b), 0≦x≦1.0,  (10)

[0088] where (w_(f)+w_(b) )/2=w₀, δ=(w_(b)−w_(f))/2w₀, a=−2δ, andb=(1+δ).

[0089] Inserting the linear width function, Equation 10, intodifferential Equation 4 allows the calculation of the deflection oftrapezoidal-shaped thermo-mechanical bending portion 63, y(x), inresponse to a load P at the free end location 18: $\begin{matrix}{{\frac{^{2}y}{x^{2}} = {\frac{12{PL}^{3}}{{Eh}^{3}w_{0}}\frac{\left( {x - 1} \right)}{\left( {{ax} + b} \right)}}},} & (11) \\{{y(x)} = {C_{0}\left\{ {{- \frac{x^{2}}{4\delta}} + {\frac{\left( {1 - \delta} \right)\left( {1 - {\left( {{2x} - 1} \right)\delta}} \right)}{8\quad \delta^{3}}\left\lbrack \quad {{- 1} - {\ln \frac{\left( {1 + \delta} \right)}{\left( {1 - {\left( {{2x} - 1} \right)\delta}} \right)}} + \frac{\left( {1 + \delta} \right)}{\left( {1 - {\left( {{2x} - 1} \right)\delta}} \right)}} \right\rbrack}} \right\}}} & (12)\end{matrix}$

[0090] where C₀ in Equation 12 above is the same constant C₀ found inEquations 7-9 for the rectangular thermo-mechanical bending portioncase. The quantity δ expresses the amount of taper in units of w₀.Further, Equation 12 above reduces to Equation 7 for the rectangularcase as δ→0.

[0091] The beneficial effects of a taper-shaped thermo-mechanicalbending portion may be further understood by examining the amount ofload P induced deflection at the free end location 18 and normalizing bythe amount of deflection, −C₀/3, that was found for the rectangularshape case (see Equation 9). The normalized deflection at the free endis designated {overscore (y)}(1): $\begin{matrix}{{\overset{\_}{y}(1)} = {{\frac{3}{4}\left\lbrack {\frac{{2\delta} - 1}{\delta^{2}} + {\frac{\left( {1 - \delta} \right)^{2}}{2\delta^{3}}\ln \frac{\left( {1 + \delta} \right)}{\left( {1 - \delta} \right)}}} \right\rbrack}.}} & (13)\end{matrix}$

[0092] The normalized free end deflection, {overscore (y)}(1), isplotted as a function of δ in FIG. 14, curve 210. Curve 210 in FIG. 14shows that as δ increases the thermo-mechanical bending portion deflectsless under the applied load P. For practical implementations, δ cannotbe increased much beyond δ=0.75 because the implied narrowing of thefree end also leads to a weak free end location 18 in the cantileveredelement 20 where the thermo-mechanical bending portion 63 joins to thefree end portion 27.

[0093] The normalized free end deflection plot 210 in FIG. 14 shows thata tapered or trapezoidal shaped thermo-mechanical bending portion willresist more efficiently an actuator load, or backpressure in the case ofa fluid moving device. For example, if a typical rectangular thermalactuator of width w₀=20 μm and length L=100 μm is narrowed at the freeend to w_(f)=10 μm, and broadened at the base end to w_(b)=30 μm, thenδ=0.5. Such a tapered thermo-mechanical bending portion will bedeflected —18% less than the 20 μm wide rectangular thermal actuatorwhich has the same area. This improved load resistance of the taperedthermo-mechanical bending portion is translated into an increase inactuation force and net free end deflection when pulsed with the sameheat energy. Alternatively, the improved force efficiency of the taperedshape may be used to provide the same amount of force while using alower energy heat pulse.

[0094] As illustrated in FIG. 12b, many shapes for the thermo-mechanicalbending portion which monotonically reduce in width from base end tofree end will show improved resistance to an actuation load orbackpressure as compared to a rectangular bender of comparable area andlength. This can be seen from Equation 4 by recognizing that the rate ofchange in the bending of the beam, d²y/dx² is caused to decrease as thewidth is increased at the base end. That is, from Equation 4:$\begin{matrix}{\frac{^{2}y}{x^{2}} \propto {\frac{\left( {1 - x} \right)}{w(x)}.}} & (14)\end{matrix}$

[0095] As compared to the rectangular case wherein w(x)=w₀, a constant,a normalized, monotonically decreasing w(x) will result in a smallernegative value for the rate of change in the slope of the beam at thebase end, which is being deflected downward under the applied load P.Therefore, the accumulated amount of beam deflection at the free end,x=1, may be less. A beneficial improvement in the thermo-mechanicalbending portion resistance to a load will be present if the base endwidth is substantially greater than the free end width, provided thefree end has not been narrowed to the point of creating a mechanicallyweak elongated structure. The term substantially greater is used hereinto mean at least 20% greater.

[0096] It is useful to the understanding of the present inventions tocharacterize thermo-mechanical bender portions that have a monotonicallyreducing width by calculating the normalized deflection at the free end,{overscore (y)}(1) subject to an applied load P, as was done above forthe linear taper shape. The normalized deflection at the free end,{overscore (y)}(1), is calculated for an arbitrary shape 62, such asthat illustrated in FIG. 12b, by first normalizing the shape parametersso that the deflection may be compared in consistent fashion to asimiliarly constructed rectangular thermo-mechanical bending portion oflength L and constant width w₀. The length of and the distance along thearbitrary shaped thermo-mechanical bender portion 62, x, are normalizedto L so that x ranges from x=0 at the anchor location 14 to x=1 at thefree end location 18.

[0097] The width reduction function, w(x), is normalized by requiringthat the average width of the arbitrary shaped thermo-mechanical benderportion 62 is w₀. That is, the normalized width reduction function,{overscore (w)}(x), is formed by adjusting the shape parameters so that$\begin{matrix}{{\int_{0}^{t}{\frac{\overset{\_}{w}(x)}{w_{0}}{x}}} = 1.} & (15)\end{matrix}$

[0098] The normalized deflection at the free end, {overscore (y)}(1), isthen calculated by first inserting the normalized width reductionfunction, {overscore (w)}(x), into differential Equation 4:$\begin{matrix}{{\frac{^{2}y}{x^{2}} = {{\frac{12{PL}^{3}}{{Eh}^{3}w_{0}}\frac{\left( {x - 1} \right)}{\overset{\_}{w}(x)}} = {C_{0}\frac{\left( {x - 1} \right)}{\overset{\_}{w}(x)}}}},} & (16)\end{matrix}$

[0099] where C₀ is the same coefficient as given in above Equation 8.

[0100] Equation 16 is integrated twice to determine the deflection,y(x), along the thermo-mechanical bender portion 62. The integrationsolutions are subjected to the boundary conditions noted above,y(0)=dy(0)/dx=0. In addition, if the normalized width reduction function{overscore (w)}(x) has steps, i.e. discontinuities, y and dy/dx arerequired to be continuous at the discontinuities. y(x) is evaluated atfree end location 18, x=1, and normalized by the quantity (−C₀/3), thefree end deflection of a rectangular thermo-mechanical bender of lengthL and width w₀. The resulting quantity is the normalized deflection atthe free end, {overscore (y)}(1): $\begin{matrix}{{\overset{\_}{y}(1)} = {{- 3}{\int_{0}^{t}{\left\lbrack {\int_{0}^{x_{2}}{\frac{\left( {x_{1} - 1} \right)}{\overset{\_}{w}\left( x_{1} \right)}{x_{1}}}} \right\rbrack {{x_{2}}.}}}}} & (17)\end{matrix}$

[0101] If the normalized deflection at the free end, {overscore(y)}(1)<1, then that thermo-mechanical bender portion shape will be moreresistant to deflection under load than a rectangular shape having thesame area. Such a shape may be used to create a thermal actuator havingmore deflection for the same input of thermal energy or the samedeflection with the input of less thermal energy than the comparablerectangular thermal actuator. If, however, {overscore (y)}(1)>1, thenthe shape is less resistant to an applied load or backpressure effectsand is disadvantaged relative to a rectangular shape.

[0102] The normalized deflection at the free end, {overscore (y)}(1), isused herein to characterize and evaluate the contribution of the shapeof the thermo-mechanical bender portion to the performance of acantilevered thermal actuator. {overscore (y)}(1) may be determined foran arbitary width reduction shape, w(x), by using well known numericalintegration methods to calculate {overscore (w)}(x) and evaluateEquation 17. All shapes which have {overscore (y)}(1)<1 are preferredembodiments of the present inventions.

[0103] Two alternative shapes which embody the present inventions areillustrated in FIG. 15. FIG. 15a illustrates a thermo-mechanical bendingportion 64 having a supralinear width reduction, in this case aquadratic change in the width from w_(b) to w_(f): $\begin{matrix}{{{w(x)} = {{\left( \frac{w_{f} - w_{b}}{L^{2}} \right)x^{2}} + w_{b}}},{0 \leq x \leq {L.}}} & (18)\end{matrix}$

[0104]FIG. 15b illustrates a stepwise reducing thermo-mechanical bendingportion 65 which has a single step reduction at x=x_(s): $\begin{matrix}\begin{matrix}{{{w(x)} = w_{b}},{0 \leq x \leq x_{s}}} \\{{= w_{f}},{x_{s} \leq x \leq {1.0.}}}\end{matrix} & (19)\end{matrix}$

[0105] A supralinear width function similar to Equation 18 will beanalyzed in closed form hereinbelow. The stepwise shape, Equation 19, ismore readily amenable to a closed form solution which further aids inunderstanding the present inventions.

[0106]FIG. 15c illustrates an alternate apparatus adapted to apply aheat pulse directly to the thermo-mechanical bending portion 65, thinfilm resistor 46. A thin film resistor may be formed on substrate 10before construction of the cantilevered element 20 and thermo-mechanicalbending portion 65, applied after completion, or at an intermediatestage. Such heat pulse application apparatus may be used with any of thethermo-mechanical bending portion designs of the present inventions.

[0107] A first stepwise reducing thermo-mechanical bending portion 65that may be examined is one that reduces at the midway point, x_(s)=0.5in units of L. That is, $\begin{matrix}\begin{matrix}{{{w(x)} = {w_{0}\left( {1 + \delta} \right)}},{0 \leq x \leq 0.5}} \\{{= {w_{0}\left( {1 - \delta} \right)}},{0.5 \leq x \leq {1.0.}}}\end{matrix} & (20)\end{matrix}$

[0108] where δ=(w_(b)−w_(f))/2w₀ and the area of the thermo-mechanicalbending portion 65 is equal to a rectangular bender of width w₀ andlength L. Equation 4 may be solved for the deflection y(x) experiencedunder a load P applied at the free end location 18 of steppedthermo-mechanical bending portion 65. The boundary conditionsy(0)=dy(0)/dx=0 are supplemented by requiring continuity in y and dy/dxat the step x_(s)=0.5. The deflection, y(x), under load P, is found tobe: $\begin{matrix}{{{{y_{1}(x)} = {\frac{C_{0}}{\left( {1 + \delta} \right)}\left\lbrack {\frac{x^{3}}{6} - \frac{x^{2}}{2}} \right\rbrack}},{0 \leq x \leq \frac{1}{2}}}{{{y_{2}(x)} = {\frac{C_{0}}{\left( {1 - \delta} \right)}\left\lbrack {\frac{x^{3}}{6} - \frac{x^{2}}{2} + {\frac{3}{4}\frac{\delta}{\left( {1 + \delta} \right)}x} - {\frac{1}{6}\frac{\delta}{\left( {1 + \delta} \right)}}} \right\rbrack}},{\frac{1}{2} \leq x \leq 1}}} & (21)\end{matrix}$

[0109] The deflection of the stepped thermo-mechanical bending portionat the free end location 18, normalized by the free end deflection ofthe rectangular bender of equal area and length is: $\begin{matrix}{{{\overset{\_}{y}}_{2}(1)} = {{\frac{1}{\left( {1 - \delta} \right)}\left\lbrack {1 - {\frac{7}{4}\frac{\delta}{\left( {1 + \delta} \right)}}} \right\rbrack}.}} & (22)\end{matrix}$

[0110] Equation 22 is plotted as plot 220 in FIG. 16 as a function of δ.It can be seen that the stepped thermo-mechanical bending portion 65shows an improved resistance to the load P for fractions up to aboutδ˜0.5 at which point the beam becomes weak and the resistance declines.By choosing a step reduction of ˜0.5 w₀, the stepped beam will deflect˜16% less than a rectangular thermo-mechanical bending portion of equalarea and length. This increased load resistance is comparable to thatfound for a trapezoidal shaped thermo-mechanical bending portion havinga taper fraction of δ=0.5 (see plot 210, FIG. 14).

[0111]FIG. 16 indicates that there is an optimum width reduction for agiven step position for stepped thermo-mechanical bending portions. Itis also the case that there may be an optimum step position, x_(s), fora given fractional width reduction of the stepped thermo-mechanicalbending portion. The following general, one-step width reduction case isanalyzed: $\begin{matrix}\begin{matrix}{{{w(x)} = {w_{b} = {{w_{0}\left( {1 - f + {fx}_{s}} \right)}/x_{s}}}},\quad {0 \leq x \leq x_{s}}} \\{{= {w_{f} = {w_{0}f}}},{x_{s} \leq x \leq {1.0.}}}\end{matrix} & (23)\end{matrix}$

[0112] where f is the fraction of the free end width compared to thenominal width w₀ for a rectangular thermo-mechanical bending portion,f=w_(f)/w₀. Equation 23 is substituted into differential Equation 4using the boundary conditions as before, y(0)=dy(0)/dx=0 and continuityin y and dy/dx at step x_(s). The normalized deflection at the free endlocation 18 is found to be: $\begin{matrix}{{\overset{\_}{y}(1)} = {{\frac{1}{f}\left\lbrack {1 + \frac{\left( {f - 1} \right)\left( {x_{s}^{3} - {3x_{s}^{2}} + {3x_{s}}} \right)}{\left( {1 - f + {f\quad x_{s}}} \right)}} \right\rbrack}.}} & (24)\end{matrix}$

[0113] The slope of Equation 24 as a function of x_(s) is examined todetermine the optimum values of x_(s) for a choice of f: $\begin{matrix}{\frac{{\overset{\_}{y}(1)}}{x_{s}} = {\frac{\left( {f - 1} \right)}{f}{\left\{ \frac{{\left( {1 - f + {f\quad x_{s}}} \right)\left( {{3x_{s}^{2}} - {6x_{s}} + 3} \right)} - {f\left( {x_{s}^{3} - {3x_{s}^{2}} + {3x_{s}}} \right)}}{\left( {1 - f + {f\quad x_{s}}} \right)^{2}} \right\}.}}} & (25)\end{matrix}$

[0114] The slope function in Equation 25 will be zero when the numeratorin the curly brackets is zero. The values of f and x_(s) which result inthe minimum value of the normalized deflection of the free end, f^(opt)and x_(s) ^(opt), are found from Equation 25 to obey the followingrelationship: $\begin{matrix}{f^{opt} = {\frac{{- 3}\left( {x_{s}^{opt} - 1} \right)^{2}}{{2\left( {x_{s}^{opt} - 1} \right)^{3}} - 1}.}} & (26)\end{matrix}$

[0115] The relationship between f^(opt) and x_(s) ^(opt) given inEquation 26 is plotted as curve 222 in FIG. 17.

[0116] The minimum value for the normalized deflection of the free end,{overscore (y)}_(min)(1), that can be realized for a given choice of thelocation of the step position, may be calculated by inserting the valueof f^(opt) into Equation 4 above. This yields an expression for theminimum value of the normalized deflection of the free end of a singlestep reduction thermo-mechanical bender portion that may be achieved:$\begin{matrix}{{{\overset{\_}{y}}_{\min}(1)} = {\frac{{4\left( {x_{s}^{opt} - 1} \right)^{7}} + {6\left( {x_{s}^{opt} - 1} \right)^{6}} + {2\left( {x_{s}^{opt} - 1} \right)^{4}} + {3\left( {x_{s}^{opt} - 1} \right)^{3}} - {2x} - 1}{{- 3}\left( {\left( {x_{s}^{opt} - 1} \right)^{3} + 1} \right)}.}} & (27)\end{matrix}$

[0117] The minimum value for the normalized deflection of the free end,{overscore (y)}_(min)(1), is plotted as a function of the location ofthe step position, x_(s), is plotted as curve 224 in FIG. 18. It may beseen from FIG. 18 that to gain at least a 10% improvement in loadresistance, over a standard rectangular shape for the thermo-mechanicalbender portion, the step position may be selected in the range isx_(s)˜0.3 to 0.84. Selection of x_(s) in this range, coupled withselecting f^(opt) according to Equation 26, allows reduction of thenormalized deflection of the free end to be below 0.9, i.e., {overscore(y)}(1)<0.9.

[0118] The normalized deflection, {overscore (y)}(1), at the free endlocation 18 expressed in Equation 24 is contour-plotted in FIG. 19 as afunction of the free end width fraction, f, and the step position x_(s).The contours in FIG. 19 are lines of constant {overscore (y)}(1),ranging from {overscore (y)}(1)=1.2 to {overscore (y)}(1)=0.85, aslabeled. Beneficial single step width reduction shapes are those thathave {overscore (y)}(1)<1.0. There are not choices for the parameters fand x_(s) that result in values of {overscore (y)}(1) much less than the{overscore (y)}(1)=0.85 contour in FIG. 19, as may also be understoodfrom FIG. 18. Those stepped width reduction shapes wherein {overscore(y)}(1)≧1.0 are not preferred embodiments of the present inventions.These shapes are conveyed by parameter choices in the lower left cornerof the plot in FIG. 19.

[0119] It may be understood from the contour plots of FIG. 19 that thereare multiple combinations of the two variables, f and x_(s), whichproduce some beneficial reduction in the deflection of the free endunder load. For example, the {overscore (y)}(1)=0.85 contour in FIG. 19illustrates that a mechanical bending portion could be constructedhaving a free end width fraction of f=0.5 with a step position of eitherx_(s)=0.45 or x_(s)=0.68.

[0120] A supralinear width reduction functional form which is amenableto closed form solution is illustrated in FIGS. 20a and 20 b.Thermo-mechanical bending portion 77 in FIG. 20a and thermo-mechanicalbending portion 78 in FIG. 20b have width reduction functions that havethe following quadratic form:

w(x)=2w ₀ [a−b(x+c)² ]=w ₀ {overscore (w)}(x)  (28)

[0121] where imposing the shape normalization requirement of Equation 15above results in the relation for the parameter “a” as a function of band c: $\begin{matrix}{a = {{\frac{1}{2}\left\lbrack {1 + {\frac{2b}{3}\left( {1 + {3c} + {3c^{2}}} \right)}} \right\rbrack}.}} & (29)\end{matrix}$

[0122] Further, in order that the free end of the thermo-mechanicalbending portion is greater than zero, c must satisfy: $\begin{matrix}{c < {{\frac{1}{2}\left\lbrack {\frac{1}{b} - \frac{4}{3}} \right\rbrack}.}} & (30)\end{matrix}$

[0123] Phantom rectangular shape 70 in FIGS. 20a and 20 b illustrates arectangular thermo-mechanical bender portion having the same length Land average width w₀ as the quadratic shapes 77 and 78.

[0124] The potentially beneficial effects of quadratic shapedthermo-mechanical bender portions 77 and 78, illustrated in FIGS. 20aand 20 b, may be understood by calculating the normalized deflection ofthe free end, {overscore (y)}(1), using Equation 17 and the boundaryconditions above noted. Inserting the expression for {overscore (w)}(x)given in Equation 28 into Equation 17 yields: $\begin{matrix}{{{\overset{\_}{y}(1)} = {{\frac{3}{4b}\left\{ {\sqrt{\frac{b}{a}}\left( {\frac{a}{b} + \left( {1 + c} \right)^{2}} \right){\ln \left\lbrack \frac{\left( {\sqrt{\frac{a}{b}} + 1 + c} \right)\left( {\sqrt{\frac{a}{b}} - c} \right)}{\left( {\sqrt{\frac{a}{b}} - 1 - c} \right)\left( {\sqrt{\frac{a}{b}} + c} \right)} \right\rbrack}} \right\}} + {\frac{3}{4b}\left\{ {{2\left( {1 + c} \right){\ln \left\lbrack \frac{\frac{a}{b} - \left( {1 + c} \right)^{2}}{\frac{a}{b} - c^{2}} \right\rbrack}} - 2} \right\}}}},} & (31)\end{matrix}$

[0125] where a is related to b and c as specified by Equation 29 and cis limited as specified by Equation 30.

[0126] The normalized deflection, {overscore (y)}(1), at the free endlocation 18 expressed in Equation 31 is contour-plotted in FIG. 21 as afunction of the parameters b and c. The contours in FIG. 21 are lines ofconstant {overscore (y)}(1), ranging from {overscore (y)}(1)=0.95 toy(1)=0.75, as labeled. Beneficial quadratic width reduction shapes arethose that have {overscore (y)}(1)<1.0. There are not choices for theparameters b and c that result in values of {overscore (y)}(1) much lessthan the {overscore (y)}(1)=0.75 contour in FIG. 21. It may beunderstood from the contour plots of FIG. 21, or from Equation 31directly, that the quadratic width reduction functional form Equation 28does not yield shapes having {overscore (y)}(1)>1.0. The parameter spacebounded by Equation 30 does not result in some shapes having long,narrow weak free end regions as may be the case for the single stepwidth reduction shapes discussed above or the inverse-power shapes to bediscussed hereinbelow.

[0127] It may be understood from the contour plots of FIG. 21 that thereare many combinations of the two parameters, b and c, which produce somebeneficial reduction in the deflection of the free end under load. Forexample, the {overscore (y)}(1)=0.80 contour in FIG. 21 illustrates thata beneficial thermo-mechanical bending portion could be constructedhaving a shape defined by Equation 28 wherein b=0.035 and c=8.0, pointQ, or wherein b=0.57 and c=0.0, point R. These two shapes are thoseillustrated in FIGS. 20a and 20 b. That is, thermo-mechanical benderportion 77 illustrated in FIG. 20a was formed according to Equation 28wherein a=3.032, b=0.035, and c=8.0, i.e. point Q in FIG. 21.Thermo-mechanical bender portion 78 illustrated in FIG. 20b was formedaccording to Equation 28 wherein a=0.69, b=0.57 and c=0.0, i.e. point Rin FIG. 21.

[0128] Another width reduction functional form, an inverse-powerfunction, which is amenable to closed form solution is illustrated inFIGS. 22a-22 c. Thermo-mechanical bending portions 72, 73, and 74 inFIGS. 22a-22 c, respectively, have width reduction functions that havethe following inverse-power form: $\begin{matrix}{{{w(x)} = {{2{w_{0}\left\lbrack \frac{a}{\left( {x + b} \right)^{n}} \right\rbrack}} = {w_{0}{\overset{\_}{w}(x)}}}},} & (32)\end{matrix}$

[0129] where n≧0, b>0. Imposing the shape normalization requirement ofEquation 15 above results in the relation for the parameter “a” as afunction of b and n: $\begin{matrix}{{{2a} = \frac{n - 1}{b^{1 - n} - \left( {1 + b} \right)^{1 - n}}},{n \neq 1},{{2a} = \frac{1}{\ln \left( \frac{1 + b}{b} \right)}},{n = 1.}} & (33)\end{matrix}$

[0130] Phantom rectangular shape 70 in FIGS. 22a-22 c illustrates arectangular thermo-mechanical bender portion having the same length Land average width w₀ as the inverse-power shapes 72, 73 and 74.

[0131] The potentially beneficial effects of inverse-power shapedthermo-mechanical bender portions, illustrated in FIGS. 22a-22 c, may beunderstood by calculating the normalized deflection of the free end,{overscore (y)}(1), using Equation 17 and the boundary conditions abovenoted. Inserting the expression for {overscore (w)}(x) given in Equation32 into Equation 17 yields: $\begin{matrix}{{{\overset{\_}{y}(1)} = {{3\left\lbrack \frac{b^{1 - n} - \left( {1 + b} \right)^{1 - n}}{n - 1} \right\rbrack} \times \left\{ {\left( \frac{\left( {1 + b} \right)^{n + 3} - {2b^{n + 2}} - {\left( {n + 2} \right)b^{n + 1}} - b^{n + 3}}{\left( {n + 1} \right)\left( {n + 2} \right)} \right) - \left( \frac{\left( {1 + b} \right)^{n + 3} - b^{n + 3}}{\left( {n + 2} \right)\left( {n + 3} \right)} \right)} \right\}}},} & (34)\end{matrix}$

[0132] where a is related to b and n as specified by Equation 33.

[0133] The normalized deflection at the free end location 18, {overscore(y)}(1) expressed in Equation 34, is contour-plotted in FIG. 23 as afunction of the parameters b and n. The contours in FIG. 23 are lines ofconstant {overscore (y)}(1), ranging from {overscore (y)}(1)=0.78 to{overscore (y)}(1)=1.2, as labeled. There are not choices for theparameters b and n that result in values of {overscore (y)}(1) much lessthan the {overscore (y)}(1)=0.78 contour in FIG. 23. Beneficialinverse-power width reduction shapes are those that have {overscore(y)}(1)<1.0.

[0134] It may be understood from the contour plots of FIG. 23 that thereare many combinations of the two parameters, b and n which produce somebeneficial reduction in the deflection of the free end under load. Forexample, the {overscore (y)}(1)=0.80 contour in FIG. 23 illustrates thata beneficial thermo-mechanical bending portion could be constructedhaving a shape defined by Equation 32 wherein b=1.75 and n=3, point S,or wherein b=1.5 and n=5, point T. These two shapes are thoseillustrated in FIGS. 22a and 22 b. That is, thermo-mechanical benderportion 72 illustrated in FIG. 22a was formed according to Equation 32wherein 2a=10.03, b=1.75, and n=3, i.e. point S in FIG. 23.Thermo-mechanical bender portion 73 illustrated in FIG. 22b was formedaccording to Equation 32 wherein 2a=23.25, b=1.5 and n=5 i.e. point T inFIG. 23.

[0135] The inverse-power shaped thermo-mechanical bender portion 74illustrated in FIG. 22c does not provide a beneficial resistance to anapplied load or backpressure as compared to a rectangular shape havingthe same area. Thermo-mechanical bender portion 74 is constructedaccording to Equation 32 wherein 2a=5.16, b=1, n=6, point V in FIG. 23.This shape has a normalized deflection at the free end value of{overscore (y)}(1)=1.1. Examination of the various width reductionfunctional forms discussed herein indicates that the thermo-mechanicalbender portion shape will be less efficient than a comparablerectangular shape if the free end region is made too long and narrow.Even though the widened base end width of such shapes improves theresistance to an applied load P, the long, narrow free end is so weakthat its deflection negates the benefit of the stiffer base region.Inverse-power width reduction shapes having {overscore (y)}(1)≧21.0 arenot preferred embodiments of the present inventions.

[0136] Several mathematical forms have been analyzed herein to assessthermomechanical bending portions having monotonically reducing widthsfrom a base end of width w_(b) to a free end of width w_(f), whereinw_(b) is substantially greater than w_(f). Many other shapes may beconstructed as combinations of the specific shapes analyzed herein.Also, shapes that are only slightly modified from the precisemathematical forms analyzed will have substantially the same performancecharacteristics in terms of resistant to an applied load. All shapes forthe thermo-mechanical bender portion which have normalized deflectionsof the free end values, {overscore (y)}(1)<1.0, are anticipated aspreferred embodiments of the present inventions.

[0137] The load force or back pressure resistance reduction whichaccompanies narrowing the free end of the thermo-mechanical bendingportion necessarily means that the base end is widened, for a constantarea and length. The wider base has the additional advantage ofproviding a wider heat transfer pathway for removing the activation heatfrom the cantilevered element. However, at some point a wider base endmay result in a less efficient thermal actuator if too much heat is lostbefore the actuator reaches an intended operating temperature.

[0138] Numerical simulations of the activation of trapezoidal shapedthermo-mechanical bending portions, as illustrated in FIG. 13, have beencarried out using device dimensions and heat pulses representative of aliquid drop emitter application. The calculations assumed uniformheating over the area of the thermo-mechanical bending portion 63. Thesimulated deflection of the free end location 18 achieved, against arepresentative fluid backpressure, is plotted as curve 230 in FIG. 24for tapered thermo-mechanical bending portions having taper angles Θ˜0⁰to 11⁰. The energy per pulse input was held constant as were the lengthsand overall areas of the thermo-mechanical bending portions havingdifferent taper angles. For the plot in FIG. 24, the deflection islarger for a device having more resistance to the back pressure load. Itmay be understood from plot 230, FIG. 24, that a taper angle in therange of 3⁰ to 10⁰ offers substantially increased deflection or energyefficiency over a rectangular thermo-mechanical bending portion havingthe same area and length. The rectangular device performance is conveyedby the Θ=0⁰ value of plot 230.

[0139] The fall-off in deflection at angles above 6° in plot 230 is dueto thermal losses from the widening base ends of the thermo-mechanicalbending portion. The more highly tapered devices do not reach theintended operating temperature because of premature loss in activationheat. An optimum taper or width reduction design preferably is selectedafter testing for such heat loss effects.

[0140] While much of the foregoing description was directed to theconfiguration and operation of a single thermal actuator or dropemitter, it should be understood that the present invention isapplicable to forming arrays and assemblies of multiple thermalactuators and drop emitter units. Also it should be understood thatthermal actuator devices according to the present invention may befabricated concurrently with other electronic components and circuits,or formed on the same substrate before or after the fabrication ofelectronic components and circuits.

[0141] From the foregoing, it will be seen that this invention is onewell adapted to obtain all of the ends and objects. The foregoingdescription of preferred embodiments of the invention has been presentedfor purposes of illustration and description. It is not intended to beexhaustive or to limit the invention to the precise form disclosed.Modification and variations are possible and will be recognized by oneskilled in the art in light of the above teachings. Such additionalembodiments fall within the spirit and scope of the appended claims.

PARTS LIST

[0142]10 substrate base element

[0143]12 liquid chamber

[0144]13 gap between cantilevered element and chamber wall

[0145]14 cantilevered element anchor location

[0146]15 thermal actuator

[0147]16 liquid chamber curved wall portion

[0148]18 free end of the thermo-mechanical bending portion

[0149]20 cantilevered element

[0150]21 passivation layer

[0151]22 first layer

[0152]23 second layer

[0153]25 heater resistor

[0154]26 cantilevered element anchor end portion

[0155]27 cantilevered element free end portion

[0156]28 liquid chamber structure, walls and cover

[0157]29 patterned sacrificial layer

[0158]30 nozzle

[0159]41 TAB lead

[0160]42 electrical input pad

[0161]43 solder bump

[0162]44 electrical input pad

[0163]46 thin film resistor

[0164]50 drop

[0165]52 vapor bubbles

[0166]60 working liquid

[0167]62 thermo-mechanical bending portion with monotonic widthreduction

[0168]63 trapezoidal shaped thermo-mechanical bending portion

[0169]64 thermo-mechanical bending portion with supralinear widthreduction

[0170]65 thermo-mechanical bending portion with stepped width reduction

[0171]66 heater resistor segments

[0172]68 current coupling device

[0173]70 comparable area rectangular thermo-mechanical bender portion

[0174]72 thermo-mechanical bending portion with inverse-power widthreduction

[0175]73 thermo-mechanical bending portion with inverse-power widthreduction

[0176]74 thermo-mechanical bending portion with inverse-power widthreduction

[0177]77 thermo-mechanical bending portion with quadratic widthreduction

[0178]78 thermo-mechanical bending portion with quadratic widthreduction

[0179]80 support structure

[0180]100 ink jet printhead

[0181]110 drop emitter unit

[0182]200 electrical pulse source

[0183]300 controller

[0184]400 image data source

[0185]500 receiver

What is claimed is:
 1. A thermal actuator for a micro-electromechanicaldevice comprising: (a) a base element; (b) a cantilevered elementincluding a thermo-mechanical bending portion extending from the baseelement and a free end portion residing in a first position, thethermo-mechanical bending portion having a base end width, w_(b),adjacent the base element and a free end width, w_(f), adjacent the freeend portion wherein the base end width is substantially greater than thefree end width; and (c) apparatus adapted to apply a heat pulse directlyto the thermo-mechanical bending portion causing the deflection of thefree end portion of the cantilevered element to a second position. 2.The thermal actuator of claim 1 wherein the thermo-mechanical bendingportion extends a length L from the base element to the free endportion, has an average width w₀, and has a normalized free enddeflection, {overscore (y)}(1), wherein {overscore (y)}(1)<1.0.
 3. Thethermal actuator of claim 2 wherein the width w(x) of thethermo-mechanical bending portion reduces from the base end width to thefree end width as a function of a normalized distance x measured fromx=0 at the base element to x=1 at length L from the base element andwherein w(x) has substantially a functional form w(x)=2w₀(a−b(x+c)²)having a=(1+2b(1+3c+3c²)/3)/2 and c<(1/b−4/3)/2.
 4. The thermal actuatorof claim 3 wherein the normalized free end deflection {overscore(y)}(1)<0.85.
 5. The thermal actuator of claim 2 wherein the width w(x)of the thermo-mechanical bending portion reduces from the base end widthto the free end width as a function of a normalized distance x measuredfrom x=0 at the base element to x=1 at length L from the base elementand wherein w(x) has substantially a functional form w(x)=2w₀a/(x+b)^(n)having 2a=(n−1)/(b^(1−n)−(1+b)^(1−n)), n≧0 and b>0.
 6. The thermalactuator of claim 5 wherein the normalized free end deflection{overscore (y)}(1)<0.85.
 7. The thermal actuator of claim 2 wherein thewidth of the thermo-mechanical bending portion reduces from the base endwidth to the free end width in at least one reduction step and the atleast one reduction step occurs at a distance L_(s) from the baseelement wherein 0.3 L≦L_(s)≦0.84 L.
 8. The thermal actuator of claim 2wherein the apparatus adapted to apply a heat pulse comprises a thinfilm resistor.
 9. The thermal actuator of claim 2 wherein thethermo-mechanical bending portion includes a first layer constructed ofa first material having a high coefficient of thermal expansion and asecond layer, attached to the first layer, constructed of a secondmaterial having a low coefficient of thermal expansion.
 10. The thermalactuator of claim 9 wherein the first material is electrically resistiveand the apparatus adapted to apply a heat pulse includes a resistiveheater formed in the first layer.
 11. The thermal actuator of claim 10wherein the first material is titanium aluminide.
 12. A liquid dropemitter comprising: (a) a chamber, formed in a substrate, filled with aliquid and having a nozzle for emitting drops of the liquid; (b) athermal actuator having a cantilevered element extending a from a wallof the chamber and a free end portion residing in a first positionproximate to the nozzle, the cantilevered element including athermo-mechanical bending portion extending from the base element to thefree end portion, the thermo-mechanical bending portion having a baseend width, w_(b), adjacent the base element and a free end width, w_(f),adjacent the free end portion wherein the base end width issubstantially greater than the free end width; and (c) apparatus adaptedto apply a heat pulse directly to the thermo-mechanical bending portioncausing a rapid deflection of the free end portion and ejection of aliquid drop.
 13. The liquid drop emitter of claim 12 wherein thethermo-mechanical bending portion extends a length L from the wall ofthe chamber to the free end portion, has an average width w₀, and has anormalized free end deflection, {overscore (y)}(1), wherein {overscore(y)}(1)<1.0.
 14. The liquid drop emitter of claim 13 wherein the widthw(x) of the thermo-mechanical bending portion reduces from the base endwidth to the free end width as a function of a normalized distance xmeasured from x=0 at the base element to x=1 at length L from the baseelement and wherein w(x) has substantially a functional form w(x)=2w₀(a−b(x+c)²) having a=(1+2b(1+3c+3c²)/3)/2 and c<(1/b−4/3)/2. 15.The liquid drop emitter of claim 14 wherein the normalized free enddeflection {overscore (y)}(1)<0.85.
 16. The liquid drop emitter of claim13 wherein the width w(x) of the thermo-mechanical bending portionreduces from the base end width to the free end width as a function of anormalized distance x measured from x=0 at the base element to x=1 atlength L from the base element and wherein w(x) has substantially afunctional form w(x)=2w₀a/(x+b)^(n) having2a=(n−1)/(b^(1−n)−(1+b)^(1−n)), n≧0, and b>0.
 17. The liquid dropemitter of claim 16 wherein the normalized free end deflection{overscore (y)}(1)<0.85.
 18. The liquid drop emitter of claim 13 whereinthe width of the thermo-mechanical bending portion reduces from the baseend width to the free end width in at least one reduction step and theat least one reduction step occurs at a distance L_(s) from the baseelement, wherein 0.3 L≦L_(s)≦0.84 L.
 19. The liquid drop emitter ofclaim 13 wherein the apparatus adapted to apply a heat pulse comprises athin film resistor.
 20. The liquid drop emitter of claim 12 wherein theliquid drop emitter is a drop-on-demand ink jet printhead and the liquidis an ink for printing image data.
 21. A liquid drop emitter comprising:(a) a chamber, formed in a substrate, filled with a liquid and having anozzle for emitting drops of the liquid; (b) a thermal actuator having acantilevered element extending a from a wall of the chamber and a freeend portion residing in a first position proximate to the nozzle, thecantilevered element including a thermo-mechanical bending portionextending from the base element to the free end portion, thethermo-mechanical bending portion including a first layer constructed ofan electrically resistive first material having a high coefficient ofthermal expansion and a second layer, attached to the first layer,constructed of a second material having a low coefficient of thermalexpansion., the thermo-mechanical bending portion having a base endwidth, w_(b), wherein the width of the thermo-mechanical bending portionreduces from the base end width to the free end width in a substantiallymonotonic function of the distance from the base element; (c) a heaterresistor formed in the first layer; (d) a pair of electrodes connectedto the heater resistor to apply an electrical pulse to cause resistiveheating of the thermo-mechanical bending portion causing a rapiddeflection of the free end portion and ejection of a liquid drop. 22.The liquid drop emitter of claim 21 wherein the thermo-mechanicalbending portion extends a length L from the wall of the chamber to thefree end portion, has an average width w₀, and has a normalized free enddeflection, {overscore (y)}(1), wherein {overscore (y)}(1)<1.0.
 23. Theliquid drop emitter of claim 22 wherein the width w(x) of thethermo-mechanical bending portion reduces from the base end width to thefree end width as a function of a normalized distance x measured fromx=0 at the base element to x=1 at length L from the base element andwherein w(x) has substantially a functional form w(x)=2w₀(a−b(x+c)²)having a=(1+2b(1+3c+3c²)/3)/2 and c<(1/b−4/3)/2.
 24. The liquid dropemitter of claim 23 wherein the normalized free end deflection{overscore (y)}(1)<0.85.
 25. The liquid drop emitter of claim 22 whereinthe width w(x) of the thermo-mechanical bending portion reduces from thebase end width to the free end width as a function of a normalizeddistance x measured from x=0 at the base element to x=1 at length L fromthe base element and wherein w(x) has substantially a functional formw(x )=2w₀a/(x+b)^(n) having 2a=(n−1)/(b^(1−n)−(1+b)^(1−n)), n≦0 and b>0.26. The liquid drop emitter of claim 25 wherein the normalized free enddeflection {overscore (y)}(1)<0.85.
 27. The liquid drop emitter of claim22 wherein the width of the thermo-mechanical bending portion reducesfrom the base end width to the free end width in at least one reductionstep and the at least one reduction step occurs at a distance L_(s) fromthe base element, wherein 0.3 L≦L_(s)≦0.84 L.
 28. The liquid dropemitter of claim 21 wherein the first material is titanium aluminide.29. The liquid drop emitter of claim 21 wherein the liquid drop emitteris a drop-on-demand ink jet printhead and the liquid is an ink forprinting image data.